The EMH from Voyager needed to access personnel files in order to appear as others (see "Renaissance Man"), but I was wondering how Odo does so. Does he merely need to observe the individual, does he need "files" (maybe something as basic as a touch, like the T-1000), or what? Surely he needs to at least observe the individual, unless the person/object was described to Odo in perfect detail. Or perhaps he could go off a picture.
It is never explained in canon.
The closest that DS9 comes to discussing Odo's shapeshifting / copying ability in serious terms is "The Begotten". A snippet from his conversation with Dr. Mora:
ODO: What are you doing?
MORA: I'm measuring its volume. It's been here a week and it's only grown seventeen percent. After three days in my lab, you were twice that size.
ODO: Well, maybe I was anxious to grow up so I could get out of there.
MORA: My point is, you've made no progress. By this time I'd already gotten you to mimic half a dozen simple forms.
The conversation is quite long, but it never goes into detail about how Odo learns to copy these "simple forms", let alone more complicated ones. It is mentioned that, at first, Dr. Mora physically forced Odo into contiainers of various shapes, and that electric shocks were used to train Odo to maintain the shapes, but it is not explained how Odo learned to copy shapes without being forced into them.
Out-of-universe, Odo was whatever the writers needed him to be, with both incredible feats and limitations. They were consistent regarding his inability to replicate humanoid faces, but were inconsistent about the general extent of his skills. How he is able to copy an eagle exactly but unable to copy a face at all is, unfortunately, never explained in-universe in any satisfactory way.